The Golden ratio, in which the ratio between the sum of two quantities and the larger one is the same as the ratio between the larger and smaller one, or
                    a        +        b            a        =          a      b        ,has been known in mathematics and the arts since at least ancient Greece. The unique positive solution to this ratio is an algebraic irrational number known as the Golden number,
  θ  =            1      +              5              2  (approximately 1.6180339887498948482 . . . ).
A digital encoding scheme for a 2 by 2 Multiple Input, Multiple Output (MIMO) antenna system, utilizing the Golden number and accordingly referred to as the Golden code, is described in papers by J. C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: A 2×2 full-rate space-time code with nonvanishing determinants,” published in the IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1432-1436, April 2005, and P. Dayal and M. K. Varanasi, “An optimal two transmit antenna space-time code and its stacked extensions,” published in the IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4348-4355, December 2005, both of which are incorporated herein by reference in their entirety.
The Golden code is a space-time code for 2×2 MIMO that is full-rate and full-diversity. It has many properties of interest in the field of wireless communications. For example, it had been shown that the Golden code achieves the optimal tradeoff between diversity gain and multiplexing gain in a slow-fading channel, as described in papers by H. Yao and G. W. Wornell, “Structured space-time block codes with optimal diversity-multiplexing tradeoff and minimum delay,” published in Proc. IEEE Globecom 2003, and by L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” published in IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073-1096, May 2003, both of which are incorporated herein by reference in their entirety. It has also been shown that the Golden code achieves the best possible coding gain for QAM and PAM types of modulation, by Dayal and Varanasi, supra. Because the Golden code does not suffer the loss of spectral efficiency with the increase of the signal constellation, as do other codes, it can be used with higher order modulations, and is thus a good choice in systems with adaptive selection of the modulation scheme. Due to its superiority in these key performance metrics, the Golden code has been included in, e.g., the IEEE 802.11 and 802.16 specifications. Furthermore, the Golden code has been generalized to other MIMO configurations such as 3×3, 4×4, and 6×6, as described in the paper by F. Oggier, G. Rekaya, J.-C. Belfiore, and E. Viterbo, “Perfect space-time block codes,” published in IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3885-3902, September 2006, incorporated herein by reference in its entirety.
For a 2×2 MIMO configuration, let s1, s2, s3, and s4 be four data symbols. The Golden code encodes these four symbols according to
                                 X          =                    ⁢                      [                                                                                x                                          1                      ,                      1                                                                                                            x                                          1                      ,                      2                                                                                                                                        x                                          2                      ,                      1                                                                                                            x                                          2                      ,                      2                                                                                            ]                                                                    =                        ⁢                                          1                                  5                                            ⁡                              [                                                                                                                                                          (                                                          1                              +                                                              ⅈ                                ⁢                                                                                                                                  ⁢                                                                  θ                                  _                                                                                                                      )                                                    ⁢                                                      s                            1                                                                          +                                                                              (                                                          θ                              -                              ⅈ                                                        )                                                    ⁢                                                      s                            2                                                                                                                                                                                                                    (                                                          1                              +                                                              ⅈ                                ⁢                                                                                                                                  ⁢                                                                  θ                                  _                                                                                                                      )                                                    ⁢                                                      s                            3                                                                          +                                                                              (                                                          θ                              -                              ⅈ                                                        )                                                    ⁢                                                      s                            4                                                                                                                                                                                                                                                        (                                                          ⅈ                              -                              θ                                                        )                                                    ⁢                                                      s                            3                                                                          +                                                                              (                                                          1                              +                                                              ⅈ                                ⁢                                                                                                                                  ⁢                                                                  θ                                  _                                                                                                                      )                                                    ⁢                                                      s                            4                                                                                                                                                                                                                    (                                                          1                              +                                                              ⅈ                                ⁢                                                                                                                                  ⁢                                θ                                                                                      )                                                    ⁢                                                      s                            1                                                                          +                                                                              (                                                                                          θ                                _                                                            -                              ⅈ                                                        )                                                    ⁢                                                      s                            2                                                                                                                                              ]                                              ,                    wherei=√{square root over (−1)}, and θ=1−θ.The coded symbol xi,j is transmitted from antenna i during the jth symbol interval.
Receivers for the Golden code are known in non-spread systems such as OFDM or TDMA in the presence of additive white Gaussian noise (AWGN). Typically, sphere decoding is used to recover the original symbols based on a reduced-complexity approximation to the maximum-likelihood (ML) decoder, as disclosed in the paper by B. Cerato, G. Masera, and E. Viterbo, “A VLSI decoder for the Golden code,” published in Proc. IEEE ICECS, pp. 549-553, December 2006, incorporated herein by reference in its entirety.
In spread spectrum systems such as CDMA, the Generalized Rake (G-Rake) receiver is effective in suppressing colored interference, as described in the paper by G. E. Bottomley, T. Ottosson, and Y. P. E. Wang, “A generalized RAKE receiver for interference suppression,” published in IEEE J. Sel. Areas Commun., vol. 18, no. 8, pp. 1536-1545, August 2000, incorporated herein by reference in its entirety. In a CDMA system, multipaths result in loss of signal orthogonality and increased self-interference. In this scenario, G-Rake can significantly improve performance by equalizing the channel. Typically, interference in a CDMA system can be modeled as a colored noise when there are few dominant interfering sources. G-Rake suppresses interference by accounting for interference temporal and spatial correlations in its combining weight formulation.
The G-Rake receiver was extended to deal with transmit diversity signals, such as the Alamouti encoded signal, as described in the paper by Y. P. E. Wang, G. E. Bottomley, and A. S. Khayrallah, “Transmit diversity and receiver performance in a WCDMA system,” published in the proceedings of IEEE Globecom 2007, Washington, D.C., USA, Nov. 26-30, 2007, incorporated herein by reference in its entirety. It was shown that the G-Rake combining weights derived based on channel coefficients with respect to a 1st transmit antenna and a 2nd transmit antenna, respectively, are used to combine the despread values from two symbol intervals. The G-Rake combined values are then used to formulate the decision variables. The transmitted symbols can then be individually detected based on the decision variables.
There exists a need in the art for a CDMA receiver solution for detecting the Golden encoded signal in a CDMA system, in the presence of colored noise.